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An integer is positive if it is greater than zero, and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: If a < b and c < d, then a + c < b + d; If a < b and 0 < c, then ac < bc
Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal to zero. Thus a non-negative number is either zero or positive.
When placed after special sets of numbers, plus and minus signs are used to indicate that only positive numbers and negative numbers are included, respectively. For example, + is the set of all positive integers and is the set of all negative integers. In these cases, a subscript 0 may also be added to clarify that 0 is included.
By the well-ordering principle, has a minimum element such that when =, the equation is false, but true for all positive integers less than . The equation is true for n = 1 {\displaystyle n=1} , so c > 1 {\displaystyle c>1} ; c − 1 {\displaystyle c-1} is a positive integer less than c {\displaystyle c} , so the equation holds for c − 1 ...
Negative-base systems can accommodate all the same numbers as standard place-value systems, but both positive and negative numbers are represented without the use of a minus sign (or, in computer representation, a sign bit); this advantage is countered by an increased complexity of arithmetic operations. The need to store the information ...
the product of a negative number—al-nāqiṣ (loss)—by a positive number—al-zāʾid (gain)—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative ...
Alternatively, since the natural numbers naturally form a subset of the integers (often denoted ), they may be referred to as the positive, or the non-negative integers, respectively. [50] To be unambiguous about whether 0 is included or not, sometimes a superscript " ∗ {\displaystyle *} " or "+" is added in the former case, and a subscript ...
For a positive integer n, p(n) is the number of distinct ways of representing n as a sum of positive integers. For the purposes of this definition, the order of the terms in the sum is irrelevant: two sums with the same terms in a different order are not considered to be distinct.
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